Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,12)(11,15,17,19)(13,16,22,20)(18,23,21,24)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,40,36,39);(9,11)(10,15)(12,19)(13,21)(14,17)(16,24)(18,22)(20,23)(26,30)(31,33)(32,38)(39,40);(1,2)(3,6)(4,5)(7,8)(11,17)(15,19)(18,21)(23,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,15,17,19)(13,20,22,16)(18,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=gi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259744);